검색
Article Search

JMB Journal of Microbiolog and Biotechnology

OPEN ACCESS eISSN 0454-8124
pISSN 1225-6951
QR Code

Article

Kyungpook Mathematical Journal 2020; 60(2): 279-288

Published online June 30, 2020

Copyright © Kyungpook Mathematical Journal.

Torsion in Homology of Dihedral Quandles of Even Order

Seung Yeop Yang

Department of Mathematics, Kyungpook National University, Daegu, 41566, Republic of Korea
e-mail : seungyeop.yang@knu.ac.kr

Received: March 4, 2020; Revised: March 30, 2020; Accepted: March 31, 2020

Abstract

Go to

Niebrzydowski and Przytycki conjectured that the torsion of rack and quandle homology of a dihedral quandle of order 2k is annihilated by k, unless k = 2t for t > 1. We partially prove this conjecture.

Keywords: dihedral quandle, rack and quandle homology, torsion subgroup

1. Introduction

Go to

Rack [7] and quandle [4] homology are (co)homology theories for self-distributive structures with axioms obtained diagrammatically from Reidemeister moves in classical knot theory. Various homological and homotopical knot invariants have been developed based on these (co)homology theories. More precisely, quandle cocycle invariants [4] are constructed with cocycles of quandle homology and quandle homotopy invariants [13, 17] are obtained using homotopy classes of maps from spheres to the geometric realizations of quandle homology. It is significant to calculate quandle (co)homology and determine explicit quandle cocycles for the study of these invariants.

The free parts of rack and quandle homology groups of finite quandles have been completely computed in [6, 9], but little is known about the torsion parts. The orders of torsion elements of rack and quandle homology were approximated in [9], and their relationship with the inner automorphism group of the quandle was discussed in [6]. Niebrzydowski and Przytycki [11, 12] have developed methods and conjectures about the higher dimensional homology of dihedral quandles. They conjectured that for a dihedral quandle Rp of order odd prime p, TorHnQ(Rp)=ℤpfn, where {fn} is a “delayed” Fibonacci sequence, i.e., fn = fn−1 + fn−3 and f1 = f2 = 0, f3 = 1. It was proved independently by Clauwens [5] and Nosaka [14]. In analogy to the result of group homology, it was conjectured [12] that for a finite quasigroup quandle, the torsion subgroups of its rack and quandle homology are annihilated by the order of the quandle. This was proved in [15] and generalized [16, 19] for some connected quandles.

In this paper, we study the torsion subgroups of rack and quandle homology of non-connected quandles. To start with, we partially solve the following conjecture:

Conjecture 1.1.([12])

The number k annihilates TorHnW(R2k), unless k = 2t, t > 1 and the number 2k is the smallest number annihilating TorHnW(R2k)for k = 2t, t > 1, where W = R, Q.

1.1. Preliminaries

Definition 1.2.([8, 10])

A quandle (X, *) is an algebraic structure with a set X and a binary operation *: X × X → X satisfying the following axioms:

  • (Right self-distributivity) (a * b) * c = (a * c) * (b * c) for any a, b, c ∈ X;

  • (Invertibility) For each b ∈ X, the right translation rb: X → X given by rb(x) = x * b is invertible;

  • (Idempotency) a * a = a for any a ∈ X.

If the binary operation satisfies right self-distributivity and invertibility, then (X, *) is call a rack. Note that the three axioms above are motivated by Reidemeister moves in knot theory, and racks and quandles can be used to construct (framed) knot invariants.

Example 1.3

Basic quandles can be obtained from groups and modules as follows:

  • An abelian group A equipped with the binary operation *: A × A → A defined by g * h = 2h − g is called a Takasaki quandle or kei. Specially, when A = ℤn, it is called a dihedral quandle and denoted by Rn. See Table 1 for example.

  • A group G with the operation g * h = hg−1h is called a core quandle.

  • A group G with the conjugate operation g * h = h−1gh is called a conjugate quandle.

  • Let M be a module over the Laurent polynomial ring ℤ[T, T−1]. A quandle M with the operation a * b = Ta + (1 − T)b is called an Alexander quandle.

A trivial quandle, a set X with the binary operation a*b = a (i.e., the operation does not depend on the choice of b), is the most elementary quandle. However, it plays an important role in the proof of Theorem 2.2 because its rack and quandle homology groups do not contain torsion subgroups.

A quandle homomorphism is a map h: X → Y between two quandles (X, *) and (Y, *′) such that h(a * b) = h(a) *′ h(b) for all a, b ∈ X. A bijective quandle homomorphism is called a quandle isomorphism, and a quandle isomorphism from a quandle X onto itself is called a quandle automorphism. Let X be a quandle. The quandle automorphism group Aut(X) of X is the group consisting of quandle automorphisms of X. Note that every right translation of a quandle is a quandle automorphism. The subgroup of Aut(X) generated by all the right translations of X is called the quandle inner automorphism group, denoted by Inn(X).

A quandle X is said to be quasigroup or Latin if every left translation, lb: X → X defined by lb(x) = b * x is invertible. A quandle X is connected if the canonical action of Inn(X) on X is transitive. Otherwise, X is said to be non-connected. Note that every quasigroup quandle is connected, but the converse does not hold in general.

Example 1.4

  • (1) The dihedral quandle Rn of order n is quasigroup (i.e., connected) if n is odd. Otherwise, it is non-connected.

  • (2) The set of all 4-cycles in the symmetric group S4 with the conjugate operation is a connected quandle, but it is not quasigroup.

  • (3) An Alexander quandle is quasigroup if and only if 1 − T is invertible.

We next review the rack and quandle homology theories.

Definition 1.5.([4, 7])

  • (1) For a given rack X, let CnR(X) be the free abelian group generated by n-tuples x = (x1, …, xn) of elements of X. We define the boundary homomorphism ∂n:CnR(X)→Cn-1R(X) by for n ≥ 2,

    ∂n(x)=∑i=2n(-1)i{(x1,…,xi-1,xi+1,…,xn)-(x1*xi,…,xi-1*xi,xi+1,…,xn)}

    and for n < 2, ∂n = 0. (CnR(X), ∂n) is called the rack chain complex of X.

  • (2) For a quandle X, define the subgroup CnD(X) of CnR(X) for n ≥ 2 generated by n-tuples (x1, …, xn) with xi = xi+1 for some i. We let CnD(X)=0 if n < 2. Then (CnD(X), ∂n) forms a sub-chain complex of (CnR(X), ∂n), called the degenerate chain complex of X.

    The quotient chain complex (CnQ(X)=CnR(X)/CnD(X),∂n′), where ∂n′ is the induced homomorphism, is called the quandle chain complex of X. Hereafter, we denote all boundary maps by ∂n.

  • (3) Let A be an abelian group. Define the chain and cochain complexes

    C*W(X;A)=C*W(X)⊗A,∂=∂⊗Id,CW*(X;A)=Hom(C*W(X),A),δ=Hom(∂,Id)

    for W=R, D, and Q. The yielded homology groups

    HnW(X;A)=Hn(C*W(X;A)) and HWn(X;A)=Hn(CW*(X;A))

    for W=R, D, and Q are called the nth rack, degenerate, and quandle homology groups and the nth rack, degenerate, and quandle cohomology groups of a rack/quandle X with coefficient group A.

The free parts of the rack and quandle homology groups of a finite quandle X, denoted by FreeHnR(X) and FreeHnQ(X), respectively, were completely computed [6, 9]. In particular, for the dihedral quandle Rm of order m we have:

FreeHnR(Rm)={ℤ,if m is odd;ℤ2n,if m is even,FreeHnQ(Rm)={ℤ,if m is odd and n=1;0,if m is odd and n>1;ℤ2,if m is even.

However, it is a bit difficult to compute the torsion parts because there are fewer methods to calculate them than the group homology theory. As for the torsion parts of the rack and quandle homology of dihedral quandles, if m is odd prime, then

TorHnQ(Rm)=ℤmfn,

where fn = fn−1 + fn−3 and f1 = f2 = 0, f3 = 1 [5, 14]. Moreover,

∣Rm∣TorHnR(Rm)=0 and ∣Rm∣TorHnQ(Rm)=0

if m is odd [14, 15]. However, little is known when m is even.

2. Annihilation Theorems for Quandle Extensions

Go to

Quandle cocycles can be used to construct extensions of quandles in a similar way to obtain extensions of groups using group cocycles. An abelian extension theory for quandles was introduced by Carter, Elhamdadi, Nikiforou, and Saito [2], and a generalization to extensions with a dynamical cocycle was defined by Andruskiewitsch and Graña [1].

Definition 2.1.([1, 3])

Let X be a quandle and S be a non-empty set. Let α: X × X → Fun(S × S, S) = SS×S be a function, so that for a, b ∈ X and s, t ∈ S we have αa,b(s, t) ∈ S. Then S × X is a quandle by the operation (s, a) * (t, b) = (αa,b(s, t), a * b), where a * b denotes the quandle operation in X, if and only if α satisfies the following conditions:

  • (1) αa,a(s, s) = s for all a ∈ X and s ∈ S,

  • (2) αa,b(−, t): S → S is a bijection for each a, b ∈ X and for each t ∈ S,

  • (3) αa*b,c(αa,b(s, t), u) = αa*c,b*c(αa,c(s, u), αb,c(t, u)) for all a, b, c ∈ X and s, t, u ∈ S.

Such a function α is called a dynamical quandle cocycle. The quandle constructed above is denoted by S×α X, and is called the extension of X by a dynamical cocycle α.

We first discuss annihilation of rack and quandle homology groups of quandle extensions using certain dynamic cocycles.

Theorem 2.2

Suppose that X is a finite quasigroup quandle and S is a non-empty set. Let α be the dynamical cocycle defined by αa,b(−, t) = IdSfor all a, b ∈ X and for all t ∈ S. Then the torsion subgroups of HnR(S×αX)and HnQ(S×αX)are annihilated by |X|.

Proof

Denote an element (s, x) of S×αX by xs. Let x=(x1s1,⋯,xnsn)∈CnR(S×αX). We define two chain maps frj,fsj:CnR(S×αX)→CnR(S×αX) by

frj(x)=∣X∣ (xjs1,…,xjsj,xj+1sj+1,…,xnsn)   for   1≤j≤n,fsj(x)=∑y∈X(ys1,…,ysj,xj+1sj+1,…,xnsn)   for   1≤j≤n.

Using the following chain homotopies Dnj and Fnj, we show that Dnj:frj≃fsj for each 1 ≤ j ≤ n and Fnj:fsj-1≃frj for each 2 ≤ j ≤ n:

Dnj(x)=∑y∈X(xjs1,…,xjsj,ysj,xj+1sj+1,…,xnsn)   for   1≤j≤n,Fnj(x)=∑y∈X(xjs1,…,xjsj-1,ysj,xjsj,xj+1sj+1,…,xnsn)   for   2≤j≤n.

Note that (s, a) * (t, b) = (s, a * b), i.e., as * bt = (a * b)s since αa,b(−, t) = IdS for any a, b ∈ X and for any s, t ∈ S.

Define face maps di(*0),di(*):CnR(X)→Cn-1R(X) of the boundary homomorphism ∂n by

di(*0)(x)=(x1,…,xi-1,xi+1,…,xn) anddi(*)(x)=(x1*xi,…,xi-1*xi,xi+1,…,xn),

i.e., ∂n=∑i=2n(-1)i(di(*0)-di(*)).

Let us first consider the chain homotopy Dnj:CnR(S×αX)→Cn+1R(S×αX).

(1) Assume that i ≤ j. The idempotent condition of a quandle implies that

di(*)Dnj(x)∑y∈X(xjs1,…,xjsi-1,xjsi+1,…,xjsj,ysj,xj+1sj+1,…,xnsn).

Note that the formula above does not depend on *, in particular (di(*0)-di(*))Dnj=0. Moreover,

Dn-1jdi(*)(x)=∑y∈X(xj+1s1,…,xj+1si-1,xj+1si+1,…,xj+1sj+1,ysj+1,xj+2sj+2,…,xnsn)

which is the same as Dn-1jdi(*0)(x), hence Dn-1j(di(*0)-di(*))=0.

(2) When j + 2 ≤ i ≤ n + 1, ∑y∈X(ysj*xi-1si-1)=∑y∈X(y*xi-1)sj=∑y∈X(ysj) by the invertibility condition of a quandle, and therefore

di(*)Dnj(x)=∑y∈X((xj*xi-1)s1,…,(xj*xi-1)sj,ysj,(xj+1*xi-1)sj+1,…,(xi-2*xi-1)si-2,xisi,⋯,xnsn).

On the other hand, if j + 1 ≤ i, then we have

Dn-1jdi(*)(x)=∑y∈X((xj*xi)s1,…,(xj*xi)sj,ysj,(xj+1*xi)sj+1,…,(xi-1*xi)si-1,xi+1si+1,⋯,xnsn),.

i.e., di+1(*0)Dnj=Dn-1jdi(*0) and di+1(*0)Dnj=Dn-1jdi(*) for j + 1 ≤ i ≤ n.

(3) Suppose that i = j + 1. Then we have

di(*0)Dnj(x)=∣X∣ (xjs1,…xjsj,xj+1sj+1,…,xnsn)=frj(x).

Note that ∑y∈X(xjsk*ysj)=∑y∈X(xj*y)sk=∑y∈X(ysk) as X is a quasigroup quandle.

Therefore, we obtain the following equality:

di(*)Dnj(x)=∑y∈X(ys1,…,ysj,xj+1sj+1,…,xnsn)=fsj(x).

By (1), (2), and (3),

∂n+1Dnj(x)+Dn-1j∂n(x)=(-1)j+1(frj(x)-fsj(x)),

hence, Dnj:frj≃fsj for each 1 ≤ j ≤ n.

We next consider the chain homotopy Fnj:CnR(S×αX)→Cn+1R(S×αX).

(4) If i ≤ j − 1, then

di(*)Fnj(x)=∑y∈X(xjs1,…,xjsi-1,xjsi+1,…,xjsj-1,ysj,xjsj,xj+1sj+1,…,xnsn),

so this formula does not depend on *, in particular (di(*0)-di(*))Fnj=0.

Moreover, if i ≤ j, then

Fn-1jdi(*)(x)=∑y∈X(xj+1s1,…,xj+1si-1,xj+1si+1,…,xj+1sj,ysj+1,xj+1sj+1,xj+2sj+2,…,xnsn)

which is the same as Fn-1jdi(*0)(x), hence Fn-1j(di(*0)-di(*))=0..

(5) Note that ∑y∈X(ysj*xi-1si-1)=∑y∈X(y*xi-1)sj=∑y∈X(ysj) by the invertibility condition of a quandle.

If i = j + 1, then (di(*0)-di(*))Fnj=0. Assume that j + 2 ≤ i ≤ n + 1.

Then

di(*)Fnj(x)=∑y∈X((xj*xi-1)s1,…,(xj*xi-1)sj-1,ysj,(xj*xi-1)sj,…,(xi-2*xi-1)si-2,xisi,⋯,xnsn).

On the other hand, if j + 1 ≤ i, then

Fn-1jdi(*)(x)=∑y∈X((xj*xi)s1,…,(xj*xi)sj-1,ysj,(xj*xi)sj,…,(xi-1*xi)si-1,xi+1si+1,⋯,xnsn).

Thus, di+1(*0)Fnj=Fn-1jdi(*0) and di+1(*)Fnj=Fn-1jdi(*) for j + 1 ≤ i ≤ n.

(6) Assume that i = j. Then we have

di(*0)Fnj(x)=∣X∣ (xjs1,…,xjsj,xj+1sj+1,…,xnsn)=frj(x).

Since X is a quasigroup quandle, ∑y∈X(xjsk*ysj)=∑y∈X(xj*y)sk=∑y∈X(ysk). Hence, we have

di(*)Fnj(x)=∑y∈X(ys1,…,ysj-1,xjsj,…,xnsn)=fsj-1(x).

By (4), (5), and (6),

∂n+1Fnj(x)+Fn-1j∂n(x)=(-1)j(frj(x)-fsj-1(x)),

thus, Fnj:fsj-1≃frjfor each 2 ≤ j ≤ n.

Finally, we obtain the following sequence of chain homotopic chain maps:

∣X∣IdCnR(S×αX)=fr1≃fs1≃fr2≃⋯≃frn-1≃fsn-1≃frn≃fsn.

Let τ(S) denote the trivial quandle with the set S, i.e., s * t = s for any s, t ∈ τ(S). Consider the chain maps p:CnR(S×αX)→CnR(τ(S)) and ϕ:CnR(τ(S))→CnR(S×αX) given by

p(x1s1,⋯,xnsn)=(s1,⋯,sn) and ϕ(s1,⋯,sn)=∑y∈X(ys1,⋯,ysn).

Clearly ϕ∘p=fsn, so that we have the same induced homomorphisms

∣X∣IdHnR(S×αX)=(fr1)*=(fsn)*=ϕ*∘p*,

where p*:HnR(S×αX)→HnR(τ(S)) and ϕ*:HnR(τ(S))→HnR(S×αX). Since τ(S) is a trivial quandle, HnR(τ(S)) has no torsion. Therefore,

∣X∣z=∣X∣IdHnR(S×αX)(z)=ϕ*(p*(z))=ϕ*(0)=0

for every z∈Tor(HnR(S×αX)) as desired.

Furthermore, since the rack homology of a quandle splits into the quandle homology and the degenerate homology [9], i.e., HnR(S×αX)=HnQ(S×αX)⊕HnD(S×αX), the torsion of HnQ(S×αX) is also annihilated by |X|.

Using Theorem 2.2, we partially prove Conjecture 1.1.

Corollary 2.3

Let R2k be the dihedral quandle of order 2k. The number k annihilates both TorHnR(R2k)and TorHnQ(R2k), if k is odd.

Proof

If k = 1, we are done because R2 is a trivial quandle and therefore HnR(R2) and HnQ(R2) have no torsion by definition.

Suppose that k > 1. Let S = {e, o} be the set with two elements. We define the dynamical cocycle α: Rk × Rk → SS×S by α[a],[b](−, t) = IdS for all [a], [b] ∈ Rk and for all t ∈ S. Note that if k is odd, then S ×α Rk ≅ R2k via the quandle isomorphism h: S×αRk → R2k defined by h(e, [m]) = [2m] and h(o, [m]) = [2m + k] for each [m] ∈ Rk. Furthermore, since k is odd, the dihedral quandle Rk is a quasigroup quandle. Therefore, Theorem 2.2 implies that the torsion subgroups of HnR(R2k)=HnR(S×αRk) and HnQ(R2k)=HnQ(S×αRk) are annihilated by k.

Table 2 contains some computational results on homology groups of dihedral quandles of small even order.

Remark 2.4

Corollary 2.3 does not hold when we replace the condition “k is odd” with “k is even” in the corollary. For example, H3Q(R8)=ℤ2⊕ℤ22⊕ℤ82.

3. Future Research

Go to

Corollary 2.3 can be used to compute rack and quandle homology groups of dihedral quandles of even order. In 2016, Takefumi Nosaka suggested the following open problem during the Knots in Hellas conference:

Problem 3.1

H3Q(R2p)=ℤ⊕ℤ⊕ℤp⊕ℤpif p is odd prime.

An open problem of whether |Inn(X)| of a finite quandle X annihilates TorHnR(X) and TorHnQ(X) for every dimension n was suggested in [16]. It is known that Inn(Rm) is isomorphic to the dihedral group of order 2m if m is odd and the dihedral group of order m if m is even. One can prove the following open problem by generalizing Corollary 2.3 in case of even k:

Problem 3.2

TorHnR(Rm)and TorHnQ(Rm)are annihilated by |Inn(Rm)| for all n.

Dihedral quandle of order 6

*024351
0042042
2420420
4204204
3315315
5153153
1531531

Homology of dihedral quandles of order 2k when k is odd

n123
HnR(R2)ℤ2ℤ4ℤ8
HnQ(R2)ℤ2ℤ2ℤ2
HnD(R2)0ℤ2ℤ6
HnR(R6)ℤ2ℤ4ℤ8⊕ℤ32
HnQ(R6)ℤ2ℤ2ℤ2⊕ℤ32
HnD(R6)0ℤ2ℤ6
HnR(R10)ℤ2ℤ4ℤ8⊕ℤ52
HnQ(R10)ℤ2ℤ2ℤ2⊕ℤ52
HnD(R10)0ℤ2ℤ6
  1. N. Andruskiewitsch, and M. Graña. From racks to pointed Hopf algebras. Adv Math., 178(2)(2003), 177-243.
    CrossRef
  2. JS. Carter, M. Elhamdadi, MA. Nikiforou, and M. Saito. Extensions of quandles and cocycle knot invariants. J Knot Theory Ramifications., 12(6)(2003), 725-738.
    CrossRef
  3. JS. Carter, A. Harris, MA. Nikiforou, and M. Saito. Cocycle knot invariants, quandle extensions, and Alexander matrices. Low-dimensional topology of tomorrow (Japanese) (Kyoto, 2002), Sūrikaisekikenkyūsho Kōkyūroku., 1272(2002), 12-35.
  4. JS. Carter, D. Jelsovsky, S. Kamada, L. Langford, and M. Saito. Quandle cohomology and state-sum invariants of knotted curves and surfaces. Trans Amer Math Soc., 355(10)(2003), 3947-3989.
    CrossRef
  5. F. Clauwens. The algebra of rack and quandle cohomology. J Knot Theory Ramifications., 20(11)(2011), 1487-1535.
    CrossRef
  6. P. Etingof, and M. Graña. On rack cohomology. J Pure Appl Algebra., 177(1)(2003), 49-59.
    CrossRef
  7. R. Fenn, C. Rourke, and B. Sanderson. Trunks and classifying spaces. Appl Categ Structures., 3(4)(1995), 321-356.
    CrossRef
  8. D. Joyce. A classifying invariant of knots, the knot quandle. J Pure Appl Algebra., 23(1)(1982), 37-65.
    CrossRef
  9. RA. Litherland, and S. Nelson. The Betti numbers of some finite racks. J Pure Appl Algebra., 178(2)(2003), 187-202.
    CrossRef
  10. SV. Matveev. Distributive groupoids in knot theory. (Russian) Mat Sb (NS)., 119(161)(1)(1982), 78-88, 160.
  11. M. Niebrzydowski, and JH. Przytycki. Homology of dihedral quandles. J Pure Appl Algebra., 213(5)(2009), 742-755.
    CrossRef
  12. M. Niebrzydowski, and JH. Przytycki. Homology operations on homology of quandles. J Algebra., 324(7)(2010), 1529-1548.
    CrossRef
  13. T. Nosaka. On homotopy groups of quandle spaces and the quandle homotopy invariant of links. Topology Appl., 158(8)(2011), 996-1011.
    CrossRef
  14. T. Nosaka. On quandle homology groups of Alexander quandles of prime order. Trans Amer Math Soc., 365(7)(2013), 3413-3436.
    CrossRef
  15. JH. Przytycki, and SY. Yang. The torsion of a finite quasigroup quandle is annihilated by its order. J Pure Appl Algebra., 219(10)(2015), 4782-4791.
    CrossRef
  16. JH. Przytycki, and SY. Yang. Annihilation of torsion in homology of finite m-AQ quandles. J Knot Theory Ramifications., 25(12)(2016) 1642012, 16 pp.
    CrossRef
  17. SY. Yang. Extended quandle spaces and shadow homotopy invariants of classical links. J Knot Theory Ramifications., 26(3)(2017) 1741010, 13 pp.
    CrossRef
  18. Yang. SY. Khovanov Homology, Distributive Structure Homology and Applications to Knot Theory, Thesis (PhD), The George Washington University, 2017.
  19. SY. Yang. On torsion in homology of finite connected quandles. to appear in J Knot Theory Ramifications., .